Question: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{q^2 - 81}{q - 9}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = q$ $ b = \sqrt{81} = -9$ So we can rewrite the expression as: $r = \dfrac{({q} {-9})({q} + {9})} {q - 9} $ We can divide the numerator and denominator by $(q - 9)$ on condition that $q \neq 9$ Therefore $r = q + 9; q \neq 9$